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A331259 Numerator of harmonic mean of 3 consecutive primes. Denominators are A331260.

%I A331259 #14 Jul 29 2024 06:20:47
%S A331259 90,315,1155,3003,7293,12597,22287,38019,62031,99789,141081,195693,
%T A331259 248583,321339,146969,572241,723399,870531,1041783,1228371,1435983,
%U A331259 1750719,2149617,2615799,3027273,3339363,3603867,3953757,4692777,5639943,6837807,7483899,8512221
%N A331259 Numerator of harmonic mean of 3 consecutive primes. Denominators are A331260.
%H A331259 Robert Israel, <a href="/A331259/b331259.txt">Table of n, a(n) for n = 1..10000</a>
%F A331259 a(n) = numerator ((3*p1*p2*p3)/(p2*p3 + p1*p3 + p1*p2)) with p1 = prime(n), p2 = prime(n + 1), p3 = prime(n + 2).
%F A331259 a(n) = p1*p2*p3 if p1 == p2 == p3 (mod 3), otherwise 3*p1*p2*p3. - _Robert Israel_, Jul 29 2024
%e A331259 b(1) = a(1)/A331260(1) = 3*2*3*5 / (3*5 + 2*5 + 2*3) = 90/31,
%e A331259 b(2) = a(2)/A331260(2) = 3*3*5*7 / (5*7 + 3*7 + 3*5) = 315/71,
%e A331259 ...
%e A331259 b(15) = a(15)/A331260(15) = 3*47*53*59 / (53*59 + 47*59 + 47*53) = 440907/8391 = 146969/2797. The common factor of 3 (see A292530) makes the denominator different from A127345(15).
%p A331259 q:= proc(a,b,c) if nops({a,b,c} mod 3) = 1 then a*b*c else 3*a*b*c fi end proc:
%p A331259 P:= [seq(ithprime(i),i=1..102)]:
%p A331259 seq(q(P[i],P[i+1],P[i+2]),i=1..100); # _Robert Israel_, Jul 29 2024
%o A331259 (PARI) hm3(x,y,z)=3/(1/x+1/y+1/z);
%o A331259 p1=2; p2=3; forprime(p3=5,150, print1(numerator(hm3(p1,p2,p3)),", ");p1=p2;p2=p3)
%Y A331259 Cf. A046301, A127345, A292530, A331260.
%K A331259 nonn,frac,look
%O A331259 1,1
%A A331259 _Hugo Pfoertner_, Jan 19 2020